Applications of Integral Calculus

Evaluate the definite integral $\int_{0}^{1} \sqrt[m]{x^{n}}\, dx$, $n>0$.

Evaluate the definite integral $\int_{0}^{1} \frac{dx}{\sqrt[m]{x^{n}}}$, $m>n$.

Evaluate the definite integral $\int_{0}^{1} \frac{dx}{x^{2}+1}$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{dx}{x^{2}+3}$.

Evaluate the definite integral $\int_{-2}^{-1} \frac{dx}{x}$.

Evaluate the definite integral $\int_{0}^{\pi /3} \tg x\, dx$.

Evaluate the definite integral $\int_{0}^{a} \frac{x\, dx}{\sqrt{a^{2}-x^{2}}}$.

Evaluate the definite integral $\int_{0}^{a} \frac{dx}{\sqrt{a^{2}-x^{2}}}$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{dx}{x^{2}+a^{2}}$.

Evaluate the definite integral $\int_{-1}^{1} \frac{dx}{x^{2}}$.

Evaluate the definite integral $\int_{-1}^{1} \frac{dx}{x^{2}-1}$.

Prove the equality

Prove the equality

Evaluate the definite integral $\int_{0}^{1} \frac{x^{2}-1}{(x+2)^{4}}\, dx$.

Evaluate the definite integral $\int_{0}^{1} \frac{x^{3}}{(x^{3}+1)^{2}}\, dx$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{x^{2}(1-x^{2})}{(1+x^{2})^{3}}\, dx$.

Evaluate the definite integral $\int_{1}^{\infty } \frac{dx}{x(1+x^{4})^{3}}$.

Evaluate the definite integral $\int_{1}^{\infty } \frac{\sqrt{1+x^{2}}}{x^{4}}\, dx$.

Evaluate the definite integral $\int_{0}^{2\pi } \frac{a+\cos x}{1+2a\cos x+a^{2}}\, dx$.

Evaluate the definite integral $\int_{0}^{1} e^{-x}\sin \pi x\, dx$.

Evaluate the definite integral $\int_{0}^{\pi } x^{2}\sin nx\, dx$, $n$ an integer.

Evaluate the definite integral $\int_{0}^{\pi /4} \sqrt{\tg x}\, dx$.

Evaluate the definite integral $\int_{-1}^{1} e^{k\arcsin x}\, dx$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{dx}{\sqrt{e^{x}+a^{2}}}$.

Evaluate the definite integral $\int_{0}^{1} \frac{x^{3}\arcsin x}{\sqrt{1-x^{2}}}\, dx$.

Prove that the area bounded by the arc of the sine curve $y=\sin x$ and the $Ox$ axis between the points $(0,0)$ and $(\pi ,0)$ equals two square units.

Prove that the area of the curvilinear trapezoid bounded below by the segment of the $Ox$ axis between the points $(N,0)$ and $(2N,0)$, on the sides by verticals, and above by the hyperbola $y=\frac{1}{x}$, equals $\ln 2$.

Prove that the area bounded by the lemniscate of Bernoulli $r^{2}=a^{2}\cos 2\varphi$ equals $a^{2}$.

Prove that the area of the segment between the parabola $y=ax^{2}$ and the line $y=h$ equals $\frac{2}{3}$ of the product of the base and the height. (Archimedes.)

Prove that the volume of a paraboloid of revolution having the same base and height as a given cylinder equals half the volume of the cylinder. (Archimedes.)

Liquid spills over the edge of a vessel in the shape of a cylinder with radius $a$ and height $h$. Prove that at the moment when half of the bottom becomes exposed, the volume of the remaining liquid equals $\frac{2}{3}a^{2}h$. (Archimedes.)

Prove that the volume of the pitcher obtained by revolving the sine curve $y=r+b\sin \frac{2\pi x}{h}$ about the axis $Ox$ for $0<x<h$ equals $\pi r^{2}h+\frac{1}{2}\pi b^{2}h$.

A rod of length $l$ rotates about one of its ends, making $n$ revolutions per second. Determine the tension at the point of attachment, given that the weight per unit length of the rod equals $\sigma$, and the centrifugal force for a mass $m$ moving in a circle of radius $r$ with angular velocity $\omega$ equals $mr\omega^{2}$.

Under a load $f$, a wire of length $l$ with cross-sectional area $s$ and Young's modulus $E$ undergoes an elongation $\Delta l$ equal to $\frac{fl}{Es}$. Determine the elongation of such a wire hanging vertically under its own weight. The specific weight of the wire's material equals $\delta$.

Under a load of $1$ kg a wire stretches by $1$ cm. What work must be done to stretch it by $4$ cm?

Find the water pressure on a vertical wall of triangular shape. The height of the triangle is $h$, and the base $a$ lies on the surface of the liquid.

The same question for a wall in the shape of a semicircle whose diameter $2a$ lies on the surface of the water.

What work must be expended to pile up a conical heap of sand with radius $1{,}2$ m and height $1$ m, if the specific weight of the sand is $2$?

Find the position of the center of gravity of a homogeneous cone.

Find the center of gravity of a hemisphere of radius $R$.

A cylinder with radius $15$ cm and height $60$ cm is filled with air under a pressure of $1$ kg/cm${}^{2}$. What work must be done during isothermal compression of the gas to half its volume?

A point moves along the axis $Ox$, starting from the point $(1,0)$, so that its speed is numerically equal to its abscissa. Where will it be after $10$ seconds from the start of the motion?

Electric charges repel one another with a force $\frac{e_{1}e_{2}}{r^{2}}$, where $e_{1}$ and $e_{2}$ are the magnitudes of the charges and $r$ is the distance between them. Determine the work required to bring a charge $e_{2}=1$ from infinity to a distance $R$ from a charge $e_{1}$.

The Earth attracts a body of mass $m$ with a force $f=mg\frac{R^{2}}{r^{2}}$, where $R$ is the radius of the Earth and $r$ is the distance of the body from the Earth's center ($r \geqslant R$). Determine the work needed to remove the body to infinity from the Earth's surface.

By Joule's law, the amount of heat released by a direct current equals $Rci^{2}t$, where $c=0{,}24$ is a constant, $R$ is the resistance, $t$ is the number of seconds, and $i$ is the current. Find the heat released for an alternating current $i=a\cos bt$.

By Torricelli's law, the speed of the outflowing liquid equals $\sqrt{2gh}$, where $h$ is the depth of the opening below the surface of the liquid. Determine the time for the liquid to flow out of a cylinder of height $h$ and cross-sectional area $s$ through an opening of area $\sigma$ in the bottom.

The same question for a conical funnel with vertex at the bottom; $s$ is the area of the base, $h$ is the height of the funnel.

A disk of thickness $h$ and radius $r$ consists of a material with specific weight $\delta$ and makes $n$ revolutions per second. What work will it perform while braking to a stop?

On maps in the Mercator projection, the scale of the image increases from the equator to the pole in such a way that the distance between two meridians remains constant. The distance between meridians drawn $10°$ apart equals $a$. Find the distance between the equator and the parallel $\varphi °$.

Prove that the area $S$ of the curvilinear trapezoid bounded by the $Ox$ axis, the lines $x=a$, $x=b$, and the cubic parabola $y=Ax^{3}+Bx^{2}+Cx+D$ can be expressed by the formula

where $y_{0}$, $y_{1}$, and $y_{2}$ are the ordinates at $x=a$, $\frac{a+b}{2}$, and $b$.

Prove that the area of the curvilinear trapezoid bounded by the $Ox$ axis, the lines $x=-h$, $x=k$, and the parabola passing through the points $(-h,y_{1})$, $(0,y_{2})$, $(k,y_{3})$, can be expressed by the formula

Prove that the area $S$ of the curvilinear trapezoid bounded by the $Ox$ axis, the lines $x=a$, $x=b$, and the parabola $y=Ax^{3}+Bx^{2}+Cx+D$ can be computed by the formula

where $y_{1}$, $y_{2}$, $y_{3}$ are the values of $y$ at $x=\frac{a+2}{2}-\frac{b-a}{2}\frac{1}{\sqrt{2}}$, $x=\frac{a+b}{2}$, $x=\frac{a+b}{2}+\frac{b-a}{2}\frac{1}{\sqrt{2}}$. (Chebyshev.)

Prove that the same area for the parabola of degree 5 $y=Ax^{5}+Bx^{4}+Cx^{3}+Dx^{2}+Ex+F$ can be computed by the formula $S=\frac{b-a}{18}(5y_{1}+8y_{2}+5y_{3})$, where $y_{1}$, $y_{2}$, $y_{3}$ are the values of $y$ at

(Gauss.)

Prove that the area $S$ of a segment of a parabola can be expressed by the formula $S=\frac{2}{3}lh$, where $l$ is the length of the chord of the segment, and $h$ is the height of the segment.

Find the area of the figure bounded by the parabolas $y=x^{2}$, $y^{2}=x$.

Find the area bounded by one arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$ and the $Ox$ axis.

Find the area bounded by the cardioid $r=a(1+\cos \varphi )$.

Find the area between $Oy$ and the curve $y=\pm \sqrt{\frac{1-x}{x}}$.

Find the area of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$, by bringing its equation to parametric form.

Find the area bounded by the curve $x^{4}+y^{4}=x^{2}+y^{2}$, by switching to polar coordinates.

By switching to polar coordinates, find the area bounded by the loop of the folium of Descartes $x^{3}+y^{3}=3axy$.

By switching to polar coordinates, find the area bounded by the lemniscate $(x^{2}+y^{2})^{2}=2a^{2}xy$.

By switching to polar coordinates, find the area bounded by the pedal curve of the ellipse $(x^{2}+y^{2})^{2}=a^{2}x^{2}+b^{2}y^{2}$.

By switching to polar coordinates, find the area bounded by the $Ox$ axis and the tractrix

Here it is convenient to introduce a parameter $\varphi$ by setting $y=a\sin \varphi$.

Find the area lying inside the circle $x^{2}+y^{2}=4px$ and the parabola $y^{2}=2px$.

Find the area between the curves $27py^{2}=8(x-p)^{3}$ and $y^{2}=2px$.

Find the area between a parabola and one of its normals.

Draw a normal to the parabola cutting off a segment of least area.

Find the area lying above the $Ox$ axis and bounded by the lines: $y^{2}=4ax$; $x^{2}+y^{2}=2ax$; $2x-y=4a$.

Find the area of the curvilinear quadrilateral bounded by the parabolas: $y^{2}-9ax+81a^{2}=0$; $y^{2}-4ax+16a^{2}=0$; $y^{2}-ax+a^{2}=0$ and the $Ox$ axis. Show that this area equals the area bounded by the chords.

Determine the area lying between the circle $x^{2}+y^{2}=a^{2}$ and its pedal curve, with respect to the interior point $(b,0)$.

The cycloid

is intersected by the line $y=ka$ (with $0<k<2$). Determine the areas bounded by these lines.

The elongated cycloid $x=a(kt-\sin t)$, $y=a-a\cos t$ (with $0<k<1$) forms loops. Determine the area of the loop. (In particular for $k=\frac{2}{\pi }$; $k=\frac{3}{5\pi }$.)

Find the area of the loop of the strophoid $(a-x)y^{2}=(a+x)x^{2}$.

Find the area bounded by the lines: $y(x^{2}+4)=4(2-x)$; $x=0$; $y=0$.

Find the area bounded by the line $(y-x)^{2}=x^{3}$ and the $Ox$ axis.

Find the area bounded by the half-wave of the curve $y=\sin^{3}x+\cos^{3}x$ and the $Ox$ axis.

Determine the area bounded by the parabola

and the coordinate axes.

Determine one of the areas between the ellipse

and the hyperbola $xy=4a^{2}$.

Determine the area between the parabola $ay=(x-3a)^{2}$ and the hyperbola $xy=4a^{2}$.

Determine the area between the $Ox$ axis, the involute of a circle

and its first vertical tangent.

Determine the area of the part of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ lying outside the circle $x^{2}+y^{2}=ab$.

Determine the area of the curvilinear square belonging to both ellipses

Show that the area between one arch of an epicycloid and the base circle equals $\pi b^{2}\left(3+2\frac{b}{a}\right)$, where $a$ and $b$ are respectively the radii of the base circle and the rolling circle.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos \varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos 2\varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos 3\varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r=a\cos 4\varphi$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: $r^{2}=a^{2}\frac{\sin 3\varphi }{\sin \varphi }$ for $0<\varphi <\frac{\pi }{3}$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: the ellipse $r=\frac{p}{1+e\cos \varphi }$, $e<1$, $0<\varphi <\varphi_{0}$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: the hyperbola $\frac{p}{1+e\cos \varphi }$, $e>1$, $0<\varphi <\varphi_{0}$; $e\cos \varphi_{0}>-1$.

By switching to polar coordinates, find the area bounded by the line given in polar coordinates: the equilateral hyperbola

By switching to polar coordinates, find the area bounded by the line

for $0<\omega <\omega_{0}$.

By switching to polar coordinates, find the area bounded by the line

for $0<\omega <\omega_{0}$.

The conchoid of Nicomedes $r=\frac{a}{\sin \varphi }\pm b$, for $a<b$, has a loop. Find the area of the loop.

Find the area bounded by the outer contour of the curve

The curve $r=a\left(\frac{\sqrt{3}}{2}+\cos \frac{\varphi }{2}\right)$ has three loops, whose areas are: $a^{2}\left(\frac{5}{4}\pi +2\sqrt{3}\right)$; $a^{2}\left(\frac{5}{6}\pi -\frac{5}{4}\sqrt{3}\right)$; $a^{2}\left(\frac{5}{12}\pi -\frac{3}{4}\sqrt{3}\right)$. Verify this.

Determine the area lying between the loops of the limaçon of Pascal $r=a+b\cos \varphi$, for $a<b$.

Find the area lying between the curves $r^{2}\cos 2\varphi =a^{2}$ and $r^{2}\cos 2\varphi =4a^{2}\cos^{4}\varphi$.

Determine the area of the lune lying inside the lemniscate $r^{2}=a^{2}\cos 2\varphi$ and outside the circle $r=\frac{a}{\sqrt{2}}$.

Determine the areas bounded by the arcs of the cardioid $r=a(1+\cos \varphi )$ and the circle $r=\frac{3}{2}a$.

Determine the area of one loop of the curve

Determine the area bounded by the curve $x^{4}+y^{4}=2a^{2}xy$.

In what ratio does the line $x+y=2a$ divide the area of the loop of the folium of Descartes $x^{3}+y^{3}=3axy$?

In what ratio does the parabola $y^{2}=ax$ divide the area of the loop of the folium of Descartes $x^{3}+y^{3}=3axy$?

Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and the ellipse obtained from it by rotating about the center through an angle $\alpha <\frac{\pi }{2}$. Determine the area occupied by all points lying inside both ellipses.

Determine the area of the lune lying inside one and outside the other ellipse of the previous problem.

Previous problem (3362): given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and the ellipse obtained from it by rotating about the center through an angle $\alpha <\frac{\pi }{2}$.

Determine the area bounded by the loop of the curve $x^{7}+y^{7}=ax^{3}y^{3}$.

Determine the area bounded by the curve

Find the arc length of the curve $y=\sqrt{x}$ for $0<x<1$.

Find the arc length of the curve $y=\ln x$ for $\sqrt{3}<x<\sqrt{8}$.

Find the arc length of the curve $y=a\ln \cos \frac{x}{a}$ for $0<x<b$.

Find the arc length of the curve $y=a\ch \frac{x}{a}$ for $0<x<x_{0}$.

Find the arc length of the curve $y=\frac{x-3a}{3}\sqrt{\frac{x}{a}}$ for $0<x<x_{0}$.

Find the arc length of the curve $y=2a\ln \frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}} - 4\sqrt{ax}$ for $0<x<x_{0}$.

Find the arc length of the curve $x=a\ln \frac{a+\sqrt{a^{2}-y^{2}}}{y} - \sqrt{a^{2}-y^{2}}$ for $b<y<a$.

Find the arc length of the curve $y^{3}=px^{2}$ for $0<y<y_{0}$.

Find the arc length of the curve $(2a-x)y^{2}=x^{3}$ for $0<x<x_{0}$, where $x_{0}<2a$.

Find the arc length of the curve $x^{2/3}+y^{2/3}=a^{2/3}$. The full perimeter.

Find the arc length of the curve $x=a\cos^{5}t$, $y=a\sin^{5}t$. The full perimeter.

Find the arc length of the curve $x=a(t-\sin t)$, $y=a(1-\cos t)$. One arch.

Find the arc length of the curve $r=a(1+\cos \varphi )$. The full perimeter.

Find the arc length of the first turn of the Archimedean spiral $r=a\varphi$.

Find the arc length of the logarithmic spiral $r=ae^{m\varphi }$, $0<r<a$.

Prove that the length of one arch of the epicycloid

equals $8a\frac{n+1}{n}$.

Prove that the length of one arch of the hypocycloid

equals $8a\frac{n-1}{n}$.

Determine the arc length of the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ between its points of contact with the coordinate axes.

Find the length of the entire curve $\left(\frac{x}{a}\right)^{2/3}+\left(\frac{y}{b}\right)^{2/3}=1$.

Find the arc length of the curve

lying above the $Ox$ axis.

The parabola $4ay=x^{2}$ rolls along the $Ox$ axis. Prove that its focus traces out the catenary $y=a\ch \frac{x}{a}$.

Prove that when the cardioid $r=a(1-\sin \varphi )$ rolls along the cycloid

its cusp traces out a straight line.

Find the arc lengths of the following space curves: $x=a\cos t$, $y=a\sin t$, $z=bt$ for $0<t<t_{0}$.

Find the arc length of the space curve $x=e^{t}\cos t$, $y=e^{t}\sin t$, $z=e^{t}$ for $-\infty <t<0$.

Find the arc length of the space curve $x=at$, $y=\sqrt{3ab}\, t^{2}$, $z=2bt^{3}$ for $0<t<t_{0}$.

Find the arc length of the space curve $x^{2}=3y$, $2xy=9z$ for $0<x<x_{0}$.

Find the arc length of the space curve $y=\arcsin \frac{x}{a}$, $z=\frac{a}{4}\ln \frac{a+x}{a-x}$ for $0<x<x_{0}$.

Find the arc length of the space curve $y^{2}=2ax-x^{2}$, $z=-a\ln \left(1-\frac{x}{2a}\right)$; $0<x<x_{0}$.

Find the arc length of the space curve $4ax=(y+z)^{2}$, $4x^{2}+3y^{2}=3z^{2}$; $0<x<x_{0}$.

Find the arc length of the space curve $(y-z)^{2}=3a(y+z)$; $9x^{2}+8y^{2}=8z^{2}$; $0<x<x_{0}$.

Find the arc length of the space curve $x^{2}+y^{2}=az$, $y=x\tg \frac{z}{a}$.

Find the arc length of the space curve $x^{2}+y^{2}+z^{2}=a^{2}$, $\sqrt{x^{2}+y^{2}}\ch \arctg \frac{y}{x}=a$.

Find the volume of the solid obtained by revolving the hyperbola $xy=a^{2}$ about the axis $Ox$ for $a<x<\infty$.

Find the volume of the solid obtained by revolving the sine curve $y=\sin x$ about the axis $Ox$ for $0<x<\pi$.

Find the volume of the solid obtained by revolving the cissoid $(2a-x)y^{2}=x^{3}$ about $Ox$ for $0<x<b<2a$.

Find the volume of the solid obtained by revolving the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ about the axis $Ox$.

Find the volume of the solid obtained by revolving the cardioid $r=a(1+\cos \varphi )$ about the polar axis.

Find the volume of the solid obtained by revolving the cycloid

about the axis $Ox$, for $0<x<2\pi a$.

Find the volume of the solid obtained by revolving the parabola $y^{2}=2px$ about the axis $Ox$ for $0<x<a$.